the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...

commutators is closed and is called the derived group or the commutator subgroup of G. Commutators are used to define nilpotent and solvable groups and the...

characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center...

{\displaystyle \textstyle \prod _{i=1}^{n}h_{i}} in H/H′, where H′ is the commutator subgroup of H. The order of the factors is irrelevant since H/H′ is abelian...

group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients...

normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup...

a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the...

G^{(2)}\triangleright \cdots ,} where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of G. These two definitions...

Commutator subgroup – Smallest normal subgroup by which the quotient is commutative Abelianization – Quotienting a group by its commutator subgroup Dihedral...

center of E and [ , ] denotes the commutator. Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group...

gh = hg. commutator subgroup The commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. complete...

groups G/Z(G) (the inner automorphism group) and G′ (the commutator subgroup) and the commutator map from G/Z(G) × G/Z(G) to G′ (taking a, b to aba−1b−1)...

_{K}(t)=1} if and only if the commutator subgroup of the knot group is perfect (i.e. equal to its own commutator subgroup). For a topologically slice knot...

ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [am, bn]...

letters and denoted by An or Alt(n). For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements...

L ( n , R ) {\displaystyle \mathrm {SL} (n,\mathbb {R} )} is the commutator subgroup of the general linear group G L ( n , R ) {\displaystyle \mathrm...

central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence...

are the commutator subgroup of GL {\displaystyle \operatorname {GL} } , and the group generated by transvections. These are both subgroups of SL {\displaystyle...

Lovász conjecture.) Cayley graphs on nilpotent groups with cyclic commutator subgroup are Hamiltonian. The flip graph of a convex polygon or equivalently...

group" already had a different meaning (a group that equals its commutator subgroup). Leinster groups give a group-theoretic way of analyzing the perfect...

unique up to isomorphism. A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group F 2 {\displaystyle...

elements have only a single rotational/translational subgroup of order 2, which is also the commutator subgroup, examples [3,3]+, [3,5]+, [3,3,3]+, [3,3,5]+....

The special linear group is also the derived group (also known as commutator subgroup) of GL ⁡ ( n , F ) {\displaystyle \operatorname {GL} (n,F)} (for...

nilpotent. Some subgroup series are defined functionally, in terms of subgroups such as the center and operations such as the commutator. These include:...

Every metacyclic group is supersolvable. The commutator subgroup of a supersolvable group is nilpotent. Subgroups and quotient groups of supersolvable groups...

that: P ∩ Ap(G) is generated by the commutator subgroups [Q, NG(Q)] where Q varies over a family C of subgroups of P The choice of the family C can be...

ring A {\displaystyle A} is the universal central extension of the commutator subgroup of the stable general linear group of A {\displaystyle A} . It is...

its quotient by its commutator subgroup. An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself. The IA automorphisms...

of a nilpotent non-abelian group. The center and the commutator subgroup of Q8 is the subgroup { e , e ¯ } {\displaystyle \{e,{\bar {e}}\}} . The inner...

two points, is not sporadic, but is an almost simple group whose commutator subgroup is the alternating group A6. It is thus related to the exceptional...